Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees

We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on differences of spin at neighboring sites, located regular tree with d+1 neighbors. first provide conditions terms relevant p-norms associated transfer operator Q ensure existence countable family proper Gibbs measures, describing localization different heights. Next we prove delocalized under natural Q. show that two can be fulfilled same time, then implies coexistence both types measures large classes including SOS-model, and heavy-tailed arising instance potentials logarithmic growth.